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On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$

Tom 238 / 2017

Arkady G. Leiderman, Vladimir G. Pestov, Artur H. Tomita Fundamenta Mathematicae 238 (2017), 79-100 MSC: Primary 22A05, 54H11; Secondary 06A06. DOI: 10.4064/fm188-9-2016 Opublikowany online: 24 February 2017


A topological group $G$ is said to have a local $\omega ^\omega $-base if the neighbourhood system at the identity admits a monotone cofinal map from the directed set $\omega ^\omega $. In particular, every metrizable group is such, but the class of groups with a local $\omega ^\omega $-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-archimedean ordered fields lead to natural families of non-metrizable groups with a local $\omega ^\omega $-base which nevertheless are Baire topological spaces.

More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $\omega ^\omega $-base admits a local $\omega ^\omega $-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base if and only if the finest uniformity of $X$ has an $\omega ^\omega $-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base provided $X$ is separable and the finest uniformity of $X$ has an $\omega ^\omega $-base.


  • Arkady G. LeidermanDepartment of Mathematics
    Ben-Gurion University of the Negev
    P.O.B. 653
    Beer Sheva, Israel
  • Vladimir G. PestovDepartment of Mathematics and Statistics
    University of Ottawa
    585 King Edward Avenue
    Ottawa, Ontario K1N 6N5, Canada
    Departamento de Matemática
    Universidade Federal de Santa Catarina
    Trindade, Florianópolis, SC, 88.040-900, Brazil
  • Artur H. TomitaInstituto de Matemática e Estatistica
    Universidade de São Paulo
    Rua do Matao, 1010
    CEP 05508-090, São Paulo, Brazil

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